The ratio of the number of ladies to that of gents at a party was 3 : 2. When 20 more gents joined the party, the ratio was reversed. The number of ladies present at the party was (C.P.O., 2006)

### Concept & Single Variable Alteration When a specific quantity is added to only one group in a ratio (gents), the quantity of the other group (ladies) remains constant. By equating the representations of the constant group across both ratios, we can easily find the multiplier. ### Step-by-Step Solution * Let the initial number of ladies and gents be $3x$ and $2x$ respectively. * $20$ more gents join the party. The new number of gents becomes $(2x + 20)$. * The number of ladies remains $3x$. * The new ratio is reversed, meaning it goes from $3 : 2$ to $2 : 3$ (Ladies : Gents). * Set up the equation: $$ \frac{3x}{2x + 20} = \frac{2}{3} $$ * Cross-multiply to solve for $x$: $$ 3(3x) = 2(2x + 20) $$ $$ 9x = 4x + 40 $$ $$ 9x - 4x = 40 $$ $$ 5x = 40 $$ $$ x = 8 $$ * The question asks for the number of ladies present. * Number of ladies = $3x = 3 \times 8 = 24$. ### Exam Strategy & Shortcut Make the constant component (Ladies) have the same ratio value in both scenarios. Initial Ratio (L : G) = $3 : 2$ Final Ratio (L : G) = $2 : 3$ To make the ladies' parts equal, multiply the first ratio by $2$ and the second by $3$. Initial Ratio = $6 : 4$ Final Ratio = $6 : 9$ Ladies are now $6$ parts in both. Gents went from $4$ parts to $9$ parts, an increase of $5$ parts. We know this increase is $20$ gents. So, $5 \text{ parts} = 20 \implies 1 \text{ part} = 4$. Ladies = $6 \text{ parts} = 6 \times 4 = 24$. ### Common Pitfall Misinterpreting "the ratio was reversed". Since the original ratio was Ladies to Gents ($3:2$), the reversed ratio is still Ladies to Gents, but now it is $2:3$. Getting this order backward will yield wrong results. ### Final Answer Therefore, the correct answer is **24**.